Antennas and their coupling characteristics for wireless power transfer via magnetic coupling

ABSTRACT

Optimizing a wireless power system by separately optimizing received power and efficiency. Either one or both of received power and/or efficiency can be optimized in a way that maintains the values to maximize transferred power.

This application claims priority from provisional application No.61/032,061, filed Feb. 27, 2008, the disclosure of which is herewithincorporated by reference.

BACKGROUND

Our previous applications and provisional applications, including, butnot limited to, U.S. patent application Ser. No. 12/018,069, filed Jan.22, 2008, entitled “Wireless Apparatus and Methods”, the disclosure ofwhich is herewith incorporated by reference, describe wireless transferof power.

The transmit and receiving antennas are preferably resonant antennas,which are substantially resonant, e.g., within 10% of resonance, 15% ofresonance, or 20% of resonance. The antenna is preferably of a smallsize to allow it to fit into a mobile, handheld device where theavailable space for the antenna may be limited.

An embodiment describes a high efficiency antenna for the specificcharacteristics and environment for the power being transmitted andreceived.

Antenna theory suggests that a highly efficient but small antenna willtypically have a narrow band of frequencies over which it will beefficient. The special antenna described herein may be particularlyuseful for this kind of power transfer.

One embodiment uses an efficient power transfer between two antennas bystoring energy in the near field of the transmitting antenna, ratherthan sending the energy into free space in the form of a travellingelectromagnetic wave. This embodiment increases the quality factor (Q)of the antennas. This can reduce radiation resistance (R_(r)) and lossresistance (R_(l))

SUMMARY

The present application describes the way in which the “antennas” orcoils interact with one another to couple wirelessly the powertherebetween.

BRIEF DESCRIPTION OF THE DRAWINGS

In the Drawings:

FIG. 1 shows a diagram of a wireless power circuit;

FIG. 2 shows an equivalent circuit;

FIG. 3 shows a diagram of inductive coupling;

FIG. 4 shows a plot of the inductive coupling; and

FIG. 5 shows geometry of an inductive coil.

DETAILED DESCRIPTION

FIG. 1 is a block diagram of an inductively coupled energy transmissionsystem between a source 100, and a load 150. The source includes a powersupply 102 with internal impedance Z_(s) 104, a series resistance R₄106, a capacitance C1 108 and inductance L1 110. The LC constant ofcapacitor 108 and inductor 110 causes oscillation at a specifiedfrequency.

The secondary 150 also includes an inductance L2 152 and capacitance C2154, preferably matched to the capacitance and inductance of theprimary. A series resistance R2 156. Output power is produced acrossterminals 160 and applied to a load ZL 165 to power that load. In thisway, the power from the source 102 is coupled to the load 165 through awireless connection shown as 120. The wireless communication is set bythe mutual inductance M.

FIG. 2 shows an equivalent circuit to the transmission system of FIG. 1.The power generator 200 has internal impedance Zs 205, and a seriesresistance R1 210. Capacitor C1 215 and inductor L1 210 form the LCconstant. A current I1 215 flows through the LC combination, which canbe visualized as an equivalent source shown as 220, with a value U1.

This source induces into a corresponding equivalent power source 230 inthe receiver, to create an induced power U2. The source 230 is in serieswith inductance L2 240, capacitance C2 242, resistance R2 244, andeventually to the load 165.

Considering these values, the equations for mutual inductance are asfollows:U ₂ =jωMI ₁U ₁ =jωMI ₂where:z _(M) =jωM

$z_{1} = {z_{s} + R_{1} + {j\left( {{\omega\; L_{1}} - \frac{1}{\omega\; C_{1}}} \right)}}$$z_{2} = {z_{L} + R_{2} + {j\left( {{\omega\; L_{21}} - \frac{1}{\omega\; C_{2}}} \right)}}$z _(s) =R _(s) +jX _(s)z _(L) =R _(L) +jX _(L)The Mesh equations are:

$\begin{matrix}{{U_{s} + U_{1} - {z_{1}I_{1}}} = 0} & \rightarrow & {I_{1} = {\left( {U_{s} + U_{1}} \right)/z_{1}}} \\{{U_{2} - {z_{2}I_{2}}} = 0} & \; & {I_{2} = {U_{2}/z_{2}}}\end{matrix}$ $\left. \begin{matrix}{I_{1} = \frac{U_{s} + {z_{M}I_{2}}}{z_{1}}} & {I_{2} = \frac{z_{M}I_{1}}{z_{2}}}\end{matrix}\rightarrow I_{2} \right. = {\frac{z_{M}\left( {U_{s} + {z_{M}I_{2}}} \right)}{z_{1}z_{2}} = {\left. \frac{z_{M}U_{s}}{{z_{1}z_{2}} - z_{M^{2}}}\rightarrow I_{1} \right. = {{\frac{z_{M}}{z_{M}} \cdot I_{2}} = \frac{z_{2}U_{s}}{{z_{1}z_{2}} - z_{M^{2}}}}}}$

where:

-   -   Source power:        P ₁ =Re{U _(s) ·I ₁ *}=U _(s) ·Re{I ₁*} for avg{U _(s)}=0    -   Power into load:        P ₂ =I ₂ ·I ₂ *Re{z _(L) }=|I ₂|² ·Re{z _(L) }=|I ₂|² ·R _(L)    -   Transfer efficiency:

$\eta = {\frac{P_{2}}{P_{1}} = \frac{{I_{2} \cdot I_{2}^{\star}}R_{L}}{U_{s}{Re}\left\{ I_{1}^{\star} \right\}}}$${I_{2} \cdot I_{2}^{\star}} = \frac{z_{M}z_{M}^{\star}U_{s}^{2}}{\left( {{z_{1}z_{2}} - z_{M^{2}}} \right)\left( {{z_{1}^{\star}z_{2}^{\star}} - z_{M^{2}}^{\star}} \right)}$${{Re}\left\{ I_{1}^{\star} \right\}} = {{Re}\left\{ \frac{z_{2}^{\star}U_{s}}{{z_{1}^{\star}z^{\star}} - z_{M^{2}}^{\star}} \right\}}$

Overall transfer Efficiency is therefore:

$\eta = {\frac{P_{2}}{P_{1}} = \frac{{U_{s}^{2} \cdot R_{L}}z_{M}z_{M}^{\star}}{\left( {{z_{1}z_{2}} - z_{M}^{2}} \right)\left( {{z_{1}^{\star}z_{2}^{\star}} - z_{M^{2}}^{\star}} \right)}}$Def.: z ¹ =z ₁ z ₂ −z _(M) ₂

$\left. \begin{matrix}{\left. \rightarrow\eta \right. = {\frac{P_{2}}{P_{1}} = {\frac{R_{L}z_{M}z_{M}^{\star}}{z^{\prime}z^{\star}{Re}\left\{ \frac{z_{2}^{\star}z^{\prime}}{z^{\prime}z^{\prime \star}} \right\}} = \frac{R_{L}z_{M}z_{M}^{\star}}{{Re}\left\{ {z_{2}^{\star} \cdot z^{\prime}} \right\}}}}} \\{= {\frac{R_{L}z_{M}z_{M}^{\star}}{{Re}\left\{ {z_{2}^{\star}\left( {{z_{1}z_{2}} - z_{M}^{2}} \right)} \right\}} = \frac{R_{L}{z_{M}}^{2}}{{Re}\left\{ {{z_{1}{z_{2}}^{2}} - {z_{2}^{\star}z_{M}^{2}}} \right\}}}}\end{matrix}\rightarrow\eta \right. = {\frac{P_{2}}{P_{1}} = \frac{R_{L}{z_{M}}^{2}}{{{{z_{2}}^{2} \cdot {Re}}\left\{ z_{1} \right\}} - {z_{M}^{2}{Re}\left\{ z_{2}^{\star} \right\}}}}$Re{z₁} = R_(s) + R₁ Re{z₂^(⋆)} = R_(L) + R₂${z_{2}}^{2} = {\left( {R_{L} + R_{2}} \right)^{2} + \left( {{\omega\; L_{2}} - \frac{1}{\omega\; C_{2}} + X_{L}} \right)^{2}}$z_(M)² = ω²M² z_(M²) = (j ω M)² = −ω²M²A Transfer efficiency equation can therefore be expressed as:

$\eta = {\frac{P_{2}}{P_{1}} = \frac{\omega^{2}{M^{2} \cdot R_{L}}}{{\left( {R_{s} + R_{n}} \right)\begin{bmatrix}{\left( {R_{L} + R_{2}} \right)^{2} +} \\\left( {{\omega\; L_{2}} - \frac{1}{\omega\; C_{2}} + X_{L}} \right)\end{bmatrix}} + {\omega^{2}{M^{2}\left( {R_{L} + R_{2}} \right)}}}}$Which reduces in special cases as follows:A) when ω=ω₀=1/√{square root over (L₂C₂)}, X_(L)=0 or where

${{\omega\; L_{2}} - \frac{1}{\omega\; C_{2}} + {X_{L}(\omega)}} = 0$$\begin{matrix}{\eta = \frac{P_{2}}{P_{1}}} \\{= {\frac{\omega_{0}^{2}M^{2}}{\left\lbrack {{\left( {R_{s} + R_{n}} \right)\left( {R_{L} + R_{2}} \right)} + {\omega_{0}^{2}M_{2}}} \right\rbrack} \cdot \frac{R_{L}}{\left( {R_{L} + R_{2}} \right)}}}\end{matrix}$B) when ω=ω₀, R_(s)=0:

$\begin{matrix}{\eta = \frac{P_{2}}{P_{1}}} \\{= \frac{\omega_{0}^{2}M^{2}R_{L}}{{R_{1}\left( {R_{L} + R_{2}} \right)}^{2} + {\omega_{0}^{2}{M^{2}\left( {R_{L} + R_{2}} \right)}}}}\end{matrix}$C) when ω=ω₀, R_(s)=0 R_(L)=R₂:

$\begin{matrix}{\eta = \frac{P_{2}}{P_{1}}} \\{= \frac{\omega_{0}^{2}M^{2}}{{4R_{1}R_{2}} + {2\omega_{0}^{2}M^{2}}}}\end{matrix}$D) when ω=ω₀, R_(s)=0 R_(L)=R₂ 2R₁R₂>>ω₀ ²M²:

$\eta = {\frac{P_{2}}{P_{1}} \cong {\frac{\omega_{0}^{2}M^{2}}{4R_{1}R_{2}}\left( {{weak}\mspace{14mu}{coupling}} \right)}}$where:Mutual inductance:M=k√{square root over (L ₁ L ₂)} where k is the coupling factorLoaded Q factors:

$Q_{1,L} = \frac{\omega\; L_{1}}{R_{s} + R_{1}}$$Q_{2,L} = \frac{\omega\; L_{2}}{R_{L} + R_{2}}$Therefore, the transfer efficiency in terms of these new definitions:A) when ω=ω₀

$\begin{matrix}{\eta = \frac{P_{2}}{P_{1}}} \\{= {\frac{k^{2} \cdot \frac{\omega_{0}{L \cdot \omega_{0}}L_{2}}{\left( {R_{s} + R_{1}} \right)\left( {R_{L} + R_{2}} \right)}}{1 + {k^{2} \cdot \frac{\omega_{0}{L \cdot \omega_{0}}L_{2}}{\left( {R_{s} + R_{1}} \right)\left( {R_{L} + R_{2}} \right)}}} \cdot \frac{R_{L}}{R_{L} + R_{2}}}} \\{\eta = {\frac{k^{2} \cdot Q_{1,L} \cdot Q_{2,L}}{1 + {k^{2} \cdot Q_{1,L} \cdot Q_{2,L}}} \cdot \frac{R_{L}}{R_{L} + R_{2}}}}\end{matrix}$C) when ω=ω₀, R_(L)=R₂, (R_(s)=0):

D) ω=ω₀, R_(L)=R₂, (R_(s)=0), 2R_(n)R₂>>ω₀ ²M²→1>>k²Q_(1,UL)Q_(2,UL)/2

$\eta = {\frac{P_{2}}{Pn} \cong {\frac{k^{2}Q_{1,{UL}}Q_{2,{UL}}}{4}\left( {{weak}\mspace{14mu}{coupling}} \right)}}$Q_(UL): Q unloaded

${Q_{1,{UL}} = \frac{\omega\; L_{1}}{R_{1}}};$$Q_{2,{UL}} = \frac{\omega\; L_{2}}{R_{2}}$This shows that the output power is a function of input voltage squared

P₂ = f(U_(s)²) + I₂ ⋅ I₂^(*)R_(L);$I_{2} = \frac{z_{M}U_{s}}{{z_{1}z_{2}} - z_{M^{2}}}$$P_{2} = {\frac{z_{M}z_{M}^{*}R_{L}}{\left( {{z_{1}z_{2}} - z_{M}^{2}} \right)\left( {{z_{1}^{*}z_{2}^{*}} - z_{M}^{2^{*}}} \right)} \cdot U_{s}^{2}}$$P_{2} = \frac{{z_{M}}^{2} \cdot R_{L} \cdot U_{s}^{2}}{{z_{1}z_{2}z_{1}^{*}z_{2}^{*}} + {z_{M}} + {{z_{M}}^{2} \cdot \left( {{z_{1}z_{2}} + {z_{1}^{\;^{*}}z_{2}^{*}}} \right)}}$$P_{2} = \frac{{z_{M}}^{2} \cdot R_{L} \cdot U_{s}^{2}}{{{z_{1}z_{2}}}^{2} + {{z_{M}}^{2}2{Re}\left\{ {z_{1}z_{2}} \right\}} + {z_{M}}^{4}}$z_(M) = jω M z_(M)^(*) = −jω M z_(M) = ω M = z_(M)z_(M)^(*)z_(M)^(*) = −ω ²M² = −z_(M)²z_(M)^(2^(*)) = −ω ²M² = z_(M)² = −z_(M)²z_(M)² ⋅ z_(M)^(2^(*)) = z_(M)⁴ z₁z₂ = z₁ ⋅ z₂z₁z₂ + z₁^(*)z₂^(*) = 2Re{z₁z₂} z₁ ⋅ z₂² = z₁² ⋅ z₂²

DEFINITIONS

z ₁ R′ ₁ +jX ₁ ;z ₂ =R′ ₂ +jX ₂|z ₁ z ₂|²=(R′ ₁ ² +X ₁ ²)(R′ ₂ ² +X ₂ ²)=R′ ₁ ² R′ ₂ ² +X ₁ ² R′ ₂ ² +X₂ ² R′ ₁ ² +X ₁ ² X ₂ ²Re{z ₁ z ₂ }=Re(R′ ₁ +jX ₁)(R′ ₂ +jX ₂)=R′ ₁ R′ ₂ +X ₁ X ₂|z _(M) |=X _(M)

$P_{2} = \frac{X_{M}^{2}{R_{1} \cdot U_{s}^{2}}}{\begin{matrix}{{R_{1}^{\prime 2}R_{2}^{\prime 2}} + {R_{1}^{\prime 2}X_{2}^{2}} + {R_{1}^{\prime 2}X_{1}^{2}} + {X_{1}^{2}X_{2}^{2}} +} \\{{2\; X_{M}^{2}R_{1}^{\prime}R_{2}^{\prime}} + {2\; X_{M}^{2}X_{1}X_{2}} + X_{M}^{4}}\end{matrix}}$$P_{2} = \frac{X_{M}^{2}{R_{L} \cdot U_{s}^{2}}}{\left( {{R_{1}^{\prime}R_{2}^{\prime}} + X_{M}^{2}} \right)^{2} + {R_{1}^{\prime 2}X_{2}^{2}} + {R_{2}^{\prime 2}X_{1}^{2}} + {X_{1}^{2}X_{2}^{2}} + {2X_{M}^{2}X_{1}X_{2}}}$

Therefore, when at or near the resonance condition:ω=ω₀=ω₂=ω₀ →X ₁=0,X ₂=0

$\begin{matrix}{P_{2} = \frac{X_{M}^{2}{R_{1} \cdot U_{s}^{2}}}{{R_{1}^{\prime 2}R_{2}^{\prime 2}} + {2\; X_{M}^{2}R_{1}^{\prime}R_{2}^{\prime}} + X_{M}^{4}}} \\{= {\frac{X_{M}^{2}R_{L}}{\left( {{R_{1}^{\prime}R_{2}^{\prime}} + X_{M}^{2}} \right)^{2}} \cdot U_{s}^{2}}} \\{P_{2} = {\frac{\omega_{0}^{2}M^{2}R_{L}}{\begin{matrix}{{\left( {R_{s} + R_{1}} \right)^{2}\left( {R_{1} + R_{2}} \right)^{2}} + {2\omega_{0}^{2}M^{2}}} \\{{\left( {R_{s} + R_{1}} \right)\left( {R_{L} + R_{2}} \right)} + {\omega_{0}^{4}M^{4}}}\end{matrix}} \cdot U_{s}^{2}}} \\{P_{2} = {\frac{\omega_{0}^{2}M^{2}R_{L}}{\left( {{\left( {R_{s} + R_{1}} \right)\left( {R_{1} + R_{2}} \right)} + {\omega_{0}^{2}M^{2}}} \right)^{2}}U_{s}^{2}}}\end{matrix}$Showing that the power transfer is inversely proportional to severalvariables, including series resistances.

Mutual inductance in terms of coupling factors and inductions:

$M = {k \cdot \sqrt{L_{1}L_{2}}}$ $\begin{matrix}{P_{2} = {\frac{\omega_{0}^{2}k^{2}L_{1}{L_{2} \cdot R_{L}}}{\left( {{\left( {R_{s} + R_{1}} \right)\left( {R_{1} + R_{2}} \right)} + {\omega_{0}^{2}k^{2}L_{1}L_{2}}} \right)^{2}} \cdot U_{s}^{2}}} \\{= {\frac{k^{2}\frac{\omega_{0}L_{1}\omega_{0}L_{2}}{\left( {R_{s} + R_{M}} \right)\left( {R_{1} + R_{2}} \right)}}{\left( {1 + {k^{2}\frac{\omega_{0}L_{1}\omega_{0}L_{2}}{\left( {R_{s} + R_{M}} \right)\left( {R_{1} + R_{2}} \right)}}} \right)^{2}} \cdot \frac{U_{s}^{2}R_{L}}{\left( {R_{s} + R_{1}} \right)\left( {R_{L} + R_{2}} \right)}}} \\{P_{2} = {\frac{k^{2} \cdot Q_{L\; 1} \cdot Q_{L\; 2}}{\left( {1 + {k^{2} \cdot Q_{L\; 1} \cdot Q_{L\; 2}}} \right)^{2}} \cdot \frac{R_{L}}{\left( {R_{s} + R_{1}} \right)\left( {R_{L} + R_{2}} \right)} \cdot U_{s}^{2}}}\end{matrix}$

The power output is proportional to the square of the input power, asdescribed above. However, there is a maximum input power beyond which nofurther output power will be produced. These values are explained below.The maximum input power P_(1max) is expressed as:

${P_{1,\max} = {\frac{U_{s}^{2}}{R_{s} + R_{{in},\min}} = {{Re}\left\{ {U_{s} \cdot I_{1}^{*}} \right\}}}};$R_(in,min): min. permissible input resistanceEfficiency relative to maximum input power:

$\begin{matrix}{\eta^{\prime} = \frac{P_{2}}{P_{1,\max}}} \\{= \frac{P_{2}\left( U_{s}^{2} \right)}{P_{1,\max}}}\end{matrix}$Under resonance condition ω=ω₁=ω₂=ω₀:

$\eta^{\prime} = \frac{\omega_{0}^{2}M^{2}{R_{L}\left( {R_{s} + R_{{in},\min}} \right)}}{\left\lbrack {{\left( {R_{s} + R_{1}} \right)\left( {R_{1} + R_{2}} \right)} + {\omega_{0}^{2}M^{2}}} \right\rbrack^{2}}$Equation for input power (P₁) under the resonance condition istherefore:

$P_{1} = {\frac{P_{2}}{\eta} = {\frac{\omega_{0}^{2}M^{2}{R_{L}\left\lbrack {{\left( {R_{s} + R_{1}} \right)\left( {R_{1} + R_{2}} \right)} + {\omega_{0}^{2}M^{2}}} \right\rbrack}\left( {R_{L} + R_{2}} \right)}{{\left\lbrack {{\left( {R_{s} + R_{2}} \right)\left( {R_{L} + R_{2}} \right)} + {\omega_{0}^{0}M^{2}}} \right\rbrack^{2} \cdot \omega_{0}^{2}}M^{2}R_{L}} \cdot U_{s}^{2}}}$$P_{1} = {\frac{R_{L} + R_{2}}{{\left( {R_{s} + R_{1}} \right)\left( {R_{L} + R_{2}} \right)} +_{0}^{2}M^{2}} \cdot U_{S}^{2}}$${{For}\mspace{14mu}\left( {R_{s} + R_{M}} \right)\left( {R_{L} + R_{2}} \right)}\operatorname{>>}{\omega_{0}^{2}{M^{2}:{P_{1} \cong \frac{U_{S}^{2}}{\left( {R_{s} + R_{1}} \right)}}}}$

The current ratio between input and induced currents can be expressed as

$\frac{I_{2}}{I_{1}} = {\frac{z_{M} \cdot U_{s} \cdot \left( {{z_{1}z_{2}} - z_{n}^{2}} \right)}{\left( {{z_{1}z_{2}} - z_{M}^{2}} \right)z_{2}U_{s}} = {\frac{z_{M}}{z_{2}} = \frac{j\;\omega\; M}{R_{L} + R_{2} + {j\left( {{\omega\; L_{2}} - \frac{1}{\omega\; C_{2}}} \right)}}}}$${{at}\mspace{14mu}\omega} = {\omega_{0} = \frac{1}{\sqrt{L_{2}C_{2}}}}$$\frac{I_{2}}{I_{1}} = {{\frac{j\;\omega\; M}{R_{1} + R_{2}}\mspace{14mu}{{avg}.\left\{ \frac{I_{2}}{I_{1}} \right\}}} = \frac{\pi}{2}}$Weak coupling: R₁+R₂>|ωM|→I₂<I₁Strong coupling: R₁+R₂<|jωM|→>I₂>I₁Input current I₁: (under resonance condition)

$I_{1} = {\frac{P_{1}}{U_{s}} = \frac{\left( {R_{1} + R_{2}} \right) \cdot U_{s}}{{\left( {R_{S} + R_{1}} \right)\left( {R_{L} + R_{2}} \right)} + {\omega_{0}^{2}M^{2}}}}$$I_{1} = {\frac{\left( {R_{L} + R_{2}} \right)}{{\left( {R_{s} + R_{1}} \right)\left( {R_{L} + R_{2}} \right)} + {\omega_{0}^{2}M^{2}}} \cdot U_{s}}$Output current I₂: (under resonance condition)

$I_{2} = {\frac{j\;\omega\; M}{{\left( {R_{s} + R_{1}} \right)\left( {R_{L} + R_{2}} \right)} + {\omega_{0}^{2}M^{2}}} \cdot U_{s}}$

Maximizing transfer efficiency and output power (P₂)

can be calculated according to the transfer efficiency equation:

$\eta = {\frac{P_{2}}{P_{1}} = \frac{\omega^{2}M^{2}R_{L}}{{\left( {R_{s} + R_{n}} \right)\left\lbrack {\left( {R_{L} + R_{2}} \right)^{2} + \underset{︸}{\left( {{\omega\; L_{2}} - \frac{1}{\omega\; C_{2}} + X_{L}} \right)^{2}}} \right\rbrack} + {\omega^{2}{M^{2}\left( {R_{L} + R_{2}} \right)}}}}$After reviewing this equation, an embodiment forms circuits that arebased on observations about the nature of how to maximize efficiency insuch a system.Conclusion 1)η(L₂,C₂,X_(L)) reaches maximum for

${{\omega\; L_{2}} - \frac{1}{\omega\; C_{2}} + X_{L}} = 0$That is, efficiency for any L, C, X at the receiver is maximum when thatequation is met. Transfer efficiency wide resonance condition:

$\eta = {{\frac{P_{2}}{P_{1}}❘_{\omega = \omega_{0}}} = {\frac{\omega_{0}^{2}M^{2}}{\left\lbrack {{\left( {\underset{︸}{R_{s}} + R_{n}} \right)\left( {R_{L} + R_{2}} \right)} + {\omega_{0}^{2}M^{2}}} \right\rbrack} \cdot \frac{R_{1}}{\left( {R_{L} + R_{2}} \right)}}}$Conclusion 2)To maximize η R_(s) should be R_(s)<<R₁That is, for maximum efficiency, the source resistance R_(s) needs to bemuch lower than the series resistance, e.g., 1/50, or 1/100^(th) or lessTransfer efficiency under resonance and weak coupling condition:

(R_(s) + R_(n))(R_(L) + R₂)>> ω₀²M²$\eta \cong \frac{\omega_{0}^{2}{M^{2} \cdot \overset{︷}{R_{L}}}}{\left( {R_{s} + R_{n}} \right)\left( {\underset{︸}{R_{L}} + R_{2}} \right)^{2}}$Maximizing η/(R_(L)):

$\frac{\mathbb{d}\eta}{\mathbb{d}R_{L}} = {{\frac{\omega_{0}^{2}M^{2}}{R_{s} + R_{1}} \cdot \frac{\left( {R_{L} + R_{2}} \right) - {2\; R_{L}}}{\left( {R_{L} + R_{2}} \right)^{3}}} = {\left. 0\rightarrow R_{L} \right. = R_{2}}}$Conclusion 3)ηreaches maximum for R_(L)=R₂ under weak coupling condition.That is, when there is weak coupling, efficiency is maximum when theresistance of the load matches the series resistance of the receiver.Transfer efficiency under resonance condition.Optimizing R_(L) to achieve max. η

${\frac{\mathbb{d}\eta}{\mathbb{d}R_{L}} = 0};{{\frac{\mathbb{d}}{\mathbb{d}R_{L}} \cdot \frac{\omega_{0}^{2}M^{2}R_{L}}{{\underset{\underset{R_{1}}{︸}}{\left( {R_{s} + R_{1}} \right)}\left( {R_{L} + R_{2}} \right)^{2}} + {\omega_{0}^{2}{M^{2}\left( {R_{L} + R_{2}} \right)}}}}\frac{u}{v}}$$\frac{{u \cdot v^{\prime}} - {v \cdot u^{\prime}}}{v^{2}} = 0$u ⋅ v^(′) − v ⋅ u^(′) = 0 u = ω₀²M² ⋅ R_(L); u^(′) = ω₀²M²v = R₁^(′)(R_(L) + R₂)² + ω₀²M²(R₁ + R₂)v^(′) − 2 R₁^(′)(R_(L) + R₂) + ω₀²M² $\begin{matrix}{{{u \cdot v^{\prime}} - {v \cdot u^{\prime}}} = {{\omega_{0}^{2}M^{2}{R_{L}\left( {{2{R_{1}^{\prime}\left( {R_{L} + R_{2}} \right)}} + {\omega_{0}^{2}M^{2}}} \right)}} -}} \\{\left( {{R_{1}^{\prime}\left( {R_{1} + R_{2}} \right)}^{2} + {\omega_{0}^{2}{M^{2}\left( {R_{L} + R_{2}} \right)}}} \right)} \\{= 0} \\{= {{2\; R_{1}^{\prime}{R_{L}\left( {R_{L} + R_{2}} \right)}} + {\omega_{0}^{2}M^{2}R_{L}} -}} \\{{R_{1}^{\prime}\left( {R_{L} + R_{2}} \right)}^{2} - {\omega_{0}^{2}{M^{2}\left( {R_{L} + R_{2}} \right)}}} \\{= 0} \\{{= {{2\; R_{1}^{\prime}R_{L}^{2}} + {2R_{1}^{\prime}R_{2}R_{L}} + {\omega_{0}^{2}M^{2}R_{L}} - {}}}\mspace{31mu}{{R_{1}^{\prime}R_{L}^{2}} + {2R_{1}^{\prime}R_{2}R_{L}} - {R_{1}^{\prime}R_{2}^{2}} - {\omega_{0}^{2}M^{2}R_{L}} - {\omega_{0}^{2}M^{2}R_{2}}}} \\{= 0} \\{= {{\left( {{1\; R_{1}^{\prime}} - R_{1}^{\prime}} \right)R_{L}^{2}} - {R_{1}^{\prime}R_{2}^{2}} - {\omega_{0}^{2}M^{2}R_{2}}}} \\{= 0}\end{matrix}$$R_{L}^{2} = \frac{{R_{1}^{\prime}R_{2}^{2}} + {\omega_{0}^{2}M^{2}R_{2}}}{R_{1}^{\prime}}$$\begin{matrix}{R_{L} = {\pm \sqrt{\frac{{\left( {R_{s} + R_{1}} \right)R_{2}^{2}} + {\omega_{0}^{2}M^{2}R^{2}}}{\left( {R_{s} + R_{1}} \right)}}}} \\{= {{\pm R_{2}} \cdot \sqrt{\frac{\left( {R_{s} + R_{1}} \right) + {\omega_{0}^{2}{M^{2}/R^{2}}}}{\left( {R_{s} + R_{1}} \right)}}}}\end{matrix}$$R_{L,{opt}} = {R_{2}\sqrt{1 + \frac{\omega_{0}^{2}M^{2}}{\left( {R_{s} + R_{1}} \right)R_{2}}}}$Weak coupling condition ω₀ ²M²<<(R_(s)+R₁)R₂R_(L,opt)≅R₂Conclusion 4)There exists an optimum R_(L)>R₂ maximizing ηOutput power P₂:

$P_{2} = \frac{X_{M}^{2}R_{1}U_{s}^{w}}{\left( {{R_{1}^{\prime}R_{2}^{\prime}} + X_{M}^{2}} \right)^{2} + {R_{1}^{\prime 2}\underset{︸}{X_{2}^{2}}} + {R_{2}^{\prime 2}\underset{︸}{X_{1}^{2}}} + \underset{︸}{X_{1}^{2}X_{2}^{2}} + {2X_{M}^{2}\underset{︸}{X_{2}X_{2}}}}$Conclusion 5)

-   -   Output power P₂ (X₁,X₂) reaches maximum for

$X_{1} = {{{\omega\; L_{1}} - \frac{1}{\omega\; C_{1}} + X_{s}} = 0}$$X_{2} = {{{\omega\; L_{2}} - \frac{1}{\omega\; C_{2}} + X_{L}} = 0}$that is, when there is a resonance condition at both the primary and thesecondary.Output power P₂ wide resonance condition:

$P_{2} = {\frac{\omega_{0}^{2}{M^{2} \cdot R_{L}}}{\left\lbrack {{\left( {\underset{︸}{P_{s}} + R_{1}} \right)\left( {R_{1} + R_{2}} \right)} + {\omega_{0}^{2}M^{2}}} \right\rbrack^{2}} \cdot U_{s}^{2}}$Conclusion 6)To maximize P₂,R_(s) should be R_(s)<<R₁Output power P₂ for the wide resonance and weak coupling condition:(R _(s) +R ₁)(R _(L) +R ₂)>>ω₀ ² M ²

$P_{2} \cong {\frac{\omega_{0}^{2}M^{2}R_{L}}{\left( {R_{s} + R_{1}} \right)^{2} + \left( {R_{L} + R_{2}} \right)^{1}} \cdot U_{s}^{2}}$

Conclusion 7)

P₂(R_(L)) reaches maximum for R_(L)=R₂ (see conclusion 3)

For each of the above, the >> or << can represent much greater, muchless, e.g., 20× or 1/20 or less, or 50× or 1/50^(th) or less or 100× or1/100^(th) or less.

The value R_(L) can also be optimized to maximize P₂:

$\frac{\mathbb{d}P_{2}}{\mathbb{d}R_{L}} = 0$$\frac{{u \cdot v^{\prime}} - {v \cdot u^{\prime}}}{v^{2}} = 0$u = ω₀²M²R_(L); u^(′) = ω₀²M² v = [(R₁^(′))(R_(L) + R₂) + ω₀²M²]²v^(′) = 2 ⋅ [R₁^(′)(R_(L) + R₂) + ω₀²M²] ⋅ R₁^(′)ω₀²M²⋅R_(L) ⋅ 2[R₁^(′)(R₁ + R₂) + ω₀²M²]R₁ − [R₁^(′)(R_(L) + R₂) + ω₀²M²]²⋅ = 02 R_(L)(R₁^(′2)R_(L) + R₁^(′2)R₂) + 1 R_(L)ω₀²M² ⋅ R₁^(′) − [R₁^(′)R_(L) + R₁^(′)R₂ + ω₀²M²]² = 0${{2R_{1}^{\prime 2}R_{L}^{2}} + + {2\omega_{0}^{2}M^{2}R_{1}^{\prime}R_{L}} - {R_{1}^{\prime 2}R_{L}^{2}} - {R_{1}^{\prime 2}R_{2}^{2}} - {\omega_{0}^{2}M^{4}} - {2R_{1}^{\prime 2}R_{2}R_{L}} - {2R_{1}^{\prime}\omega_{0}^{2}M^{2}R_{L}} - {2R_{1}^{\prime}R_{2}\omega_{0}^{2}M^{2}}} = {0 = {{{\left( {{2R_{1}^{\prime 2}} - R_{1}^{\prime 2}} \right)R_{L}^{2}} - {R_{1}^{\prime 2}R_{2}^{2}} - {2\; R_{1}^{\prime}R_{2}\omega_{0}^{2}M^{2}} - {\omega_{0}^{2}M^{4}}} = {0 = {{{R_{1}^{\prime 2} \cdot R_{L}^{2}} - \left( {{R_{1}^{\prime}R_{2}} + {\omega_{0}^{2}M^{2}}} \right)^{2}} = {{0R_{L}^{2}} = {{\frac{\left( {{R_{1}^{\prime}R_{2}} + {\omega_{0}^{2}M^{2}}} \right)^{2}}{R_{1}^{\prime 2}}\begin{matrix}{R_{L,{opt}} = \frac{{R_{1}^{\prime}R_{2}} + {\omega_{0}^{2}M^{2}}}{R_{1}^{\prime}}} \\{= {R_{2}\left( {1 + \frac{\omega_{0}^{2}M^{2}}{\left( {R_{s} + R_{1}} \right)R_{2}}} \right)}}\end{matrix}R_{L,{opt}}} = {{R_{2} \cdot \left( {1 + \frac{\omega_{0}^{2}M^{2}}{\left( {R_{s} + R_{1}} \right)R_{2}}} \right)}{Weak}\mspace{14mu}{coupling}\text{:}R_{L,{opt}}\begin{matrix} \cong \\ > \end{matrix}R_{2}}}}}}}}$Conclusion 8)There exists an optimum R_(L)>R₂ maximizing P₂. This R_(1opt) differsfrom the R_(1,opt) maximizing η.One embodiment operates by optimizing one or more of these values, toform an optimum value.

Inductive coupling is shown with reference to FIGS. 3, 4 FIG. 5illustrates the Inductance of a multi-turn circular loop coil

$R_{m} = \frac{R_{0} + R_{1}}{2}$

Wheeler formula (empirical)$L = \frac{0.8\;{R_{m}^{2} \cdot N^{2}}}{{6\; R_{m}} + {9\; w} + {10\left( \left( {R_{o} - R_{1}} \right) \right.}}$[Wheeler, H. A., “Simple inductance formulas for radio coils”. Proc. IREVol 16, pp. 1328-1400, October 1928.] Note: this i accurate if all threeterms in denominator are about equal. Conversion to H, m units: [L]μH$L = \frac{0.8 \cdot R_{m}^{2} \cdot \wp^{2} \cdot N^{1} \cdot 10^{- 6}}{{6\;{R_{m} \cdot \wp}} + {9 \cdot w \cdot \wp} + {10\left( {R_{0} - R_{1}} \right)\wp}}$$L = \frac{0.8 \cdot R_{m}^{2} \cdot \wp^{2} \cdot N^{2} \cdot 10^{- 6}}{{6\; R_{m}} + {9\; w} + {10\left( {R_{0} - R_{1}} \right)}}$[R_(m), R_(i), R₀, ω] = inch${1\mspace{11mu} m} = {\frac{\overset{\wp}{\overset{︷}{10}}00}{- 154} \cdot {inch}}$1H = 10⁶ μH [L] = H [R_(m)R₀R₁ω] = m

In standard form:

${L = \frac{\mu_{0} \cdot A_{m} \cdot N^{2}}{K_{c}}};$A_(m) = π ⋅ R_(m)² μ₀ = 4 π ⋅ 10⁻⁷$L = \frac{0.8 \cdot \cdot 10^{- 6} \cdot {\overset{︷}{\pi\; R}}_{m}^{2} \cdot N^{2} \cdot \overset{︷}{4\;{\pi \cdot 10^{- 7}}}}{{\pi \cdot 4}\;{\pi \cdot 10^{- 7}}\left( {{6\; R_{m}} + {9\; w} + {10\left( {R_{0} - R_{1}} \right)}} \right)}$$L = {\frac{\mu_{0}{A_{m} \cdot N^{2} \cdot 0.8}{\cdot 10}}{4\;{\pi^{2} \cdot}\left( {{6\; R_{m}} + {9\; w} + {10\left( {R_{0} - R_{1}} \right)}} \right)}\frac{1}{K_{c}}}$$R_{m} = \sqrt{\frac{A_{m}}{\pi}}$$K_{c} = \frac{{\pi^{2} \cdot 25.4}\left( {{6\sqrt{\frac{A_{m}}{\pi}}} + {9\; w} + {10\left( {R_{0} - R_{1}} \right)}} \right)}{2 \cdot 1000}$$K_{c} \cong {\frac{1}{8} \cdot \left( {{6\sqrt{\frac{A_{m}}{\pi}}} + {9\; w} + {10\left( {R_{0} - R_{1}} \right)}} \right)}$${L = \frac{\mu_{0}A_{m}N^{2}}{K_{c}}};$$A_{m} = {{\left( \frac{\left( {R_{0} + R_{1}} \right)}{2} \right)^{2} \cdot {\pi\lbrack L\rbrack}} = H}$

The inductance of a single-turn circular loop is given as:

$K_{c} = \frac{R_{m} \cdot \pi}{\left\lbrack {\frac{8\; R_{m}}{6} - 2} \right\rbrack}$${L = \frac{\mu_{0}A_{m}}{K_{c}}};{A_{m} = {{R_{m}^{2} \cdot {\pi\lbrack L\rbrack}} = H}}$where:

R_(m): mean radius in m

b: wire radius in m,

For a Numerical example:

R₁=0.13 m

R₀=0.14 m

ω=0.01 m

N=36

→L=0.746 mH

The measured inductanceL _(meas)=0.085 mH

The model fraction of Wheeler formula for inductors of similar geometry,e.g, with similar radius and width ratios is:

$K_{c} = {\frac{1}{8}\left( {{5\sqrt{\frac{A_{m}}{\pi}}} + {9\; w} + {10\left( {R_{0} - R_{1}} \right)}} \right)}$$D = \sqrt{W^{2} + \left( {R_{0} - R_{1}} \right)^{2}}$$R_{m} = \frac{R_{0} + R_{1}}{2}$

Using a known formula from Goddam, V. R., which is valid forw>(R ₀ −R ₁)

$L = {{0.03193 \cdot R_{m} \cdot N^{2}}\left\lfloor {{2.303\left( {1 + \frac{w^{2}}{32\; R_{m}^{2}} + \frac{D^{2}}{96\; R_{m}^{2}}} \right){\log\left( \frac{8\; R_{m}}{D} \right)}} -} \right\rfloor}$1w H,m units:

$L = {\mu_{0}{R_{m} \cdot {N^{2}\left\lbrack {{\left( {1 + \frac{w^{2}}{32\; R_{m}^{2}} + \frac{D^{2}}{96\; R_{m}^{2}}} \right){n\left( \frac{8\; R_{m}}{D} \right)}} -} \right\rbrack}}}$

Example 1

R₁ = 0.13 m R₀ = 0.14 m W = 0.01 m N = 36 L = 757 μH${{Ratio}\text{:}\mspace{11mu}\frac{W}{R_{0} - R_{1}}} = 1$ → y₁ =0.8483    y₂ = 0.816 From [Terman, F.]

Example 2 Given in [Goddam, V. R.]

R₀ = 8.175 inches R₁ = 7.875 inches W = 2 inches N = 57 y₁ = 0.6310 y₂ =0.142 → L = 2.5 mH (2.36 mH) ${Ratio}\text{:}\mspace{11mu}\begin{matrix}{\frac{2}{R_{0} - R_{1}} = {\frac{2}{0.3} = {6.667\mspace{14mu}{or}}}} \\{{\frac{R_{0} - R_{1}}{W} = {\frac{0.3}{2} = 0.15}}\mspace{45mu}}\end{matrix}$

where Goddam, V. R. is the Thesis Masters Louisiana State University,2005, and Terman, F. is the Radio Engineers Handbook, McGraw Hill, 1943.

Any of these values can be used to optimize wireless power transferbetween a source and receiver.

From the above, it can be seen that there are really two differentfeatures to consider and optimize in wireless transfer circuits. A firstfeature relates to the way in which efficiency of power transfer isoptimized. A second feature relates to maximizing the received amount ofpower—independent of the efficiency.

One embodiment, determines both maximum efficiency, and maximum receivedpower, and determines which one to use, and/or how to balance betweenthe two.

In one embodiment, rules are set. For example, the rules may specify:

Rule 1—Maximize efficiency, unless power transfer will be less than 1watt. If so, increase power transfer at cost of less efficiency.

Rule 2—Maximize power transfer, unless efficiency becomes less than 30%.

Any of these rules may be used as design rules, or as rules to varyparameters of the circuit during its operation. In one embodiment, thecircuit values are adaptively changes based on operational parameters.This may use variable components, such as variable resistors,capacitors, inductors, and/or FPGAs for variation in circuit values.

Although only a few embodiments have been disclosed in detail above,other embodiments are possible and the inventors intend these to beencompassed within this specification. The specification describesspecific examples to accomplish a more general goal that may beaccomplished in another way. This disclosure is intended to beexemplary, and the claims are intended to cover any modification oralternative which might be predictable to a person having ordinary skillin the art. For example, other sizes, materials and connections can beused. Other structures can be used to receive the magnetic field. Ingeneral, an electric field can be used in place of the magnetic field,as the primary coupling mechanism. Other kinds of antennas can be used.Also, the inventors intend that only those claims which use the-words“means for” are intended to be interpreted under 35 USC 112, sixthparagraph. Moreover, no limitations from the specification are intendedto be read into any claims, unless those limitations are expresslyincluded in the claims.

Where a specific numerical value is mentioned herein, it should beconsidered that the value may be increased or decreased by 20%, whilestill staying within the teachings of the present application, unlesssome different range is specifically mentioned. Where a specifiedlogical sense is used, the opposite logical sense is also intended to beencompassed.

1. A method for power transfer in a wireless power system, comprising:first optimizing efficiency of power transfer between a transmitter anda receiver of wireless power using a variable circuit component; andseparate from said first optimizing efficiency, second optimizing areceived power in said receiver.
 2. The method as in claim 1, whereinsaid first optimizing and said second optimizing are done according torules that specify information about at least one of an efficiency levelor an amount of received power.
 3. The method as in claim 2, whereinsaid information comprises information about a threshold efficiency forpower transfer.
 4. The method as in claim 2, wherein said informationcomprises information about a threshold power amount.
 5. The method asin claim 1, wherein said first optimizing or said second optimizingcomprises optimizing based on at least one of comprising a firstresonant frequency of the transmitter with a second resonant frequencyof the receiver or based on a strength of a coupling between saidtransmitter and said receiver.
 6. The method as in claim 5, wherein saidfirst optimizing or said second optimizing is performed differently whenthe transmitter is weakly coupled with the receiver as compared to whenthe transmitter is strongly coupled with the receiver.
 7. The method asin claim 5, further comprising maintaining the first resonant frequencyof the transmitter substantially equal to the second resonant frequencyof the receiver.
 8. The method as in claim 5, wherein said firstoptimizing or said second optimizing further comprises maintaining aresistance of an inductor in the receiver substantially equal to aseries resistance.
 9. The method as in claim 5, wherein said firstoptimizing or said second optimizing further comprises maintaining asource resistance at a transmitter less than a series resistance of thetransmitter.
 10. The method as in claim 1, wherein first optimizingefficiency comprises maximizing efficiency, provided that the receivedpower is greater than or equal to a threshold power amount.
 11. Themethod as in claim 1, wherein second optimizing received power comprisesmaximizing the received power, provided that the efficiency of the powertransfer is greater than or equal to a threshold efficiency.
 12. Awireless power receiver system, comprising: an inductor having aninductance value; a capacitor electrically connected to the inductor andhaving a capacitance value; and terminal connections applied to a load,wherein the inductance value and the capacitance value are optimizedaccording to a first optimization of receive power efficiency of powerreceived from a wireless power transmitter, or a second optimization ofan amount of the received power.
 13. The system as in claim 12, whereinsaid first optimization and second optimization are done according torules that specify information about at least one of an efficiency levelor the amount of received power.
 14. The system as in claim 13, whereinsaid information comprises information about a threshold efficiency forpower transfer.
 15. The system as in claim 13, wherein said informationcomprises information about a threshold power amount.
 16. The system asin claim 12, further comprising an optimizing circuit configured toperform said first optimization or said second optimization comprisesoptimizing based on at least one of comparing a first resonant frequencyof the wireless power transmitter with a second resonant frequency of areceive circuit including the inductor electrically connected to thecapacitor or based on a strength of a coupling with the wireless powertransmitter.
 17. The system as in claim 16, wherein the optimizingcircuit is configured to perform said first optimizing or said secondoptimization is differently when the wireless power transmitter isweakly couple with the receive circuit as compared to when the wirelesspower transmitter is strongly coupled with the receive circuit.
 18. Thesystem as in claim 16, wherein the optimizing circuit is configured tomaintain the first resonant frequency of the wireless power transmittersubstantially equal to the second resonant frequency of the receivecircuit.
 19. The system as in claim 16, wherein the optimizing circuitis configured to maintain an impedance of the inductor substantiallyequal to a series resistance.
 20. The system as in claim 16, wherein theoptimization circuit is further configured to maintain a sourceresistance at the wireless power transmitter as less than a seriesresistance of the wireless power transmitter.
 21. The system as in claim16, wherein the optimizing circuit comprises at least one of a firstcomponent configured to vary the inductance value of the inductor, asecond component configured to vary the capacitance value of thecapacitor, a variable resistor, or an FPGA.
 22. The system as in claim12, wherein the second optimization of an amount of received powercomprises maximizing received power, provided that the receive powerefficiency is greater than or equal to a threshold efficiency.
 23. Thesystem as in claim 12, wherein the first optimization of receive powerefficiency comprises maximizing the receive power efficiency, providedthat the amount of the received power is greater than or equal to athreshold power amount.
 24. A method of transferring wireless power,comprising: optimizing efficiency of power transfer from a transmitterto a receiver of wireless power using a variable circuit componentaccording to rules that specify separately, information about both anefficiency level and an amount of power received by the receiver. 25.The method as in claim 24, wherein said information comprisesinformation about a threshold efficiency for power transfer.
 26. Themethod as in claim 24, wherein said information comprises informationabout a threshold power amount.
 27. The method as in claim 24, whereinsaid optimizing further comprises maintaining a resistance of aninductor in the receiver substantially equal to a series resistance. 28.The method as in claim 24, wherein said optimizing further comprisesmaintaining a source resistance at a transmitter as less than a seriesresistance of the transmitter.
 29. The method as in claim 24, whereinoptimizing efficiency comprises maximizing efficiency, provided that theamount of power received is greater than or equal to a threshold poweramount.
 30. A wireless power transmitter system, comprising: an inductorhaving an inductance value; a capacitor electrically connected to theinductor and having a capacitance value; and terminal connectionsapplied to a load wherein the inductance value and the capacitance valueare optimized according to a first optimization of power transferefficiency of power transmitted to a wireless power receiver a secondoptimization of an amount of power transmitted.
 31. The system as inclaim 30, wherein said first optimization and said second optimizationare done according to rules that specify information about at least oneof an efficiency level or the amount of transmitted power.
 32. Thesystem as in claim 31, wherein said information comprises informationabout a threshold efficiency for power transfer.
 33. The system as inclaim 31, wherein said information comprises information about athreshold power transmission amount.
 34. The system as in claim 30,further comprising an optimizing circuit configured to perform saidfirst optimization or said second optimization comprises optimizingbased on at least one of comparing a first resonant frequency of atransmit circuit including the inductor and the capacitor with a secondresonant frequency of the wireless power receiver, or based on astrength of a coupling with the wireless power receiver.
 35. The systemas in claim 34, wherein the optimizing circuit is configured to performsaid first optimization or said second optimization is differently whenthe transmit circuit is weakly coupled with the wireless power receiveras compared to when the transmit circuit is strongly coupled with thewireless power receiver.
 36. The system as in claim 34, wherein theoptimizing circuit is configured to maintain the first resonantfrequency of the transmit circuit substantially equal to the secondresonant frequency of the wireless power receiver.
 37. The system as inclaim 34, wherein the optimizing circuit is configured to maintain animpedance of the inductor substantially equal to a series resistance.38. The system as in claim 34, wherein the optimizing circuit isconfigured to maintain a source resistance as less than a seriesresistance.
 39. The system as in claim 34, wherein the optimizingcircuit comprises at least one of a first component configured to varythe inductance value of the inductor, a second component configured tovary the capacitance value of the capacitor, a variable resistor, or anFPGA.
 40. The system as in claim 30, wherein the second optimization ofthe amount of power transmitted comprises maximizing the amount of powertransmitted, provided that the power transfer efficiency is greater thanor equal to a threshold efficiency.
 41. The system as in claim 30,wherein the first optimization of power transfer efficiency comprisesmaximizing the power transfer efficiency, provided that the amount ofpower transmitted is greater than or equal to a threshold power amount.42. An apparatus configured to receive wireless power comprising:receive circuitry comprising: an inductor having an inductance value; acapacitor electrically connected to the inductor and having acapacitance value; and terminal connections to a load, the receivecircuitry being configured to optimize efficiency of power transfer froma transmitter according to rules that specify separately, informationabout both an efficiency level and an amount of power received by thereceive circuitry.
 43. The apparatus as in claim 42, wherein saidreceive circuitry is configured to optimize efficiency by maximizingefficiency of the power transfer, provided that the amount of powerreceived is greater than or equal to a threshold power amount.
 44. Theapparatus as in claim 42, wherein said receive circuitry is furtherconfigured to at least one of maintain an impedance of the inductorsubstantially equal to a series resistance and maintain a resonancecondition.
 45. The apparatus as in claim 42, wherein the inductorcomprises a variable inductor, wherein the capacitor comprises avariable capacitor, and wherein the receive circuitry further comprisesat least one of a variable resistor or an FPGA.
 46. The apparatusconfigured to receive wireless power comprising: means for optimizingefficiency of power transfer from a transmitter according to rules thatspecify separately, information about an efficiency level and an amountof power received; and means for electrically connecting the wirelesspower received to a load.
 47. The apparatus of claim 46, wherein saidmeans for optimizing efficiency comprises means for maximizingefficiency of the power transfer, provided that the amount of powerreceived is greater than or equal to a threshold power amount.
 48. Theapparatus as in claim 46, further comprising at least one of means formaintaining a resistance of an inductor substantially equal to a seriesresistance or means for tuning a receive circuit to resonate in responseto the wireless power.
 49. A method for wireless power transfer,comprising: optimizing power received from a transmitter at a receiverusing a variable circuit component according to rules that specifyseparately, information about both an efficiency level of power transferfrom the transmitter and an amount of power received by the receiver.50. The method as in claim 49, wherein optimizing the power receivedcomprises maximizing the amount of power received, provided that theefficiency level is greater than or equal to a threshold efficiency. 51.The method as in claim 49, further comprising at least one ofmaintaining a resistance of an inductor substantially equal to a seriesresistance or tuning the receiver to resonate in response to the powerreceived.
 52. An apparatus configured to receive wireless power,comprising: means for optimizing power received from the transmitteraccording to rules that specify separately, information about both anefficiency level and an amount of power received; and means forconnecting the received power to a load.
 53. The apparatus as in claim52, wherein said means for optimizing comprises means for maximizing theamount of power received, provided that the efficiency level is greaterthan or equal to a threshold efficiency.
 54. The apparatus as in claim52, further comprising at least one of means for maintaining aresistance of an inductor substantially equal to a series resistance ormeans for tuning a receive circuit to resonant in response to the powerreceived.
 55. An apparatus configured to receive wireless powercomprising: receive circuitry comprising: an inductor having aninductance value; a capacitor electrically connected to the inductor andhaving a capacitance value; and terminal connections to a load, thereceive circuitry being configured to optimize power received from atransmitter according to rules that specify separately, informationabout both an efficiency level and an amount of power received by thereceive circuitry.
 56. The apparatus as in claim 55, wherein saidreceive circuitry is configured to optimize the power received bymaximizing the amount of power received, provided that the efficiencylevel is greater than or equal to a threshold efficiency.
 57. Theapparatus as in claim 55, wherein said receive circuitry is furtherconfigured to at least one of maintain an impedance of the inductorsubstantially equal to a series resistance or tune the receive circuitryto resonate in response to the power received.
 58. The apparatus as inclaim 55, wherein the inductor comprises a variable inductor, whereinthe capacitor comprises a variable capacitor, and wherein the receivecircuitry further comprises at least one of a variable resistor or anFPGA.
 59. A system for wireless power transfer, comprising: means foroptimizing power transfer between a transmitter and a receiver byoptimizing efficiency of the power received from the transmitter and byoptimizing an amount of power received by the receiver; and means forconnecting the power received to a load.
 60. The system as in claim 59,wherein optimizing efficiency and optimizing the amount of power aredone according to rules that specify information about an efficiencylevel and the amount of power received.
 61. The system as in claim 59,wherein optimizing the amount of power received comprises maximizing thepower received, provided that the efficiency level is greater than orequal to a threshold efficiency.
 62. The system as in claim 59, whereinoptimizing efficiency comprises maximizing the efficiency of the powerreceived, provided that the amount of power received is greater than orequal to a threshold power amount.